AI 中文总结
本文将一般度量空间上的码表示为邻近图独立集，提出适用于任意有限度量空间码的Gilbert-Varshamov界通用框架，引入两类核心图，推导密度阈值与码长上界，发现汉明情形下仅局部子图统计量无法超越该界，需依赖空间大规模结构性质。
AI 中文摘要
本文通过将一般度量空间上的码表示为邻近图中的独立集，研究编码理论与极值组合学的关系。我们提出适用于任意有限度量空间上码的Gilbert-Varshamov（GV）界的通用框架，并探讨全局组合参数可迫使码超出该界存在的条件。分析的核心是引入Ramsey-Sidorenko图与强制独立集图，我们确定了各类图族的密度阈值，并利用Karush-Kuhn-Tucker条件分析汉明情形下的熵优化。此外，我们通过顶点传递图与非边传递图中的分数填充推导码长的上界。研究结果表明，仅局部子图统计量不足以在汉明情形下超越GV界，说明改进必须源于空间的大规模结构性质。
英文摘要
This paper investigates the relationship between coding theory and extremal combinatorics by representing codes in general metric spaces as independent sets in proximity graphs. We provide a generalized framework for the Gilbert-Varshamov (GV) bound applicable to codes over any finite metric space and explore the conditions under which global combinatorial parameters can force the existence of codes exceeding this bound. Central to our analysis is the introduction of Ramsey-Sidorenko and independence-forcing graphs. We establish density thresholds for various graph families and utilize the Karush--Kuhn--Tucker conditions to analyze entropy optimization in the Hamming case. Furthermore, we derive upper bounds on code sizes using fractional packings in vertex-transitive and nonedge-transitive graphs. Our findings demonstrate that local subgraph statistics alone are insufficient to surpass the GV bound in the Hamming case, suggesting that improvements must stem from large-scale structural properties of the space.
Comments23 pages, no figures
